An explicit construction for $n$-contact curves to a smooth cubic via divisions and Zariski tuples

Let $E$ be a smooth cubic. A plane curve $D$ is said to be an $n$-contact curve to $E$ if the intersection multiplicities at each intersection point between $E$ and $D$ is $n$. In this note, we give an algorithm to produce $n$-contact curves to $E$ and consider its application.

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